C₂²×#46414 #72324856

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#46414

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

113113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

(6,8)(10,13)(14,16)(17,18,20,21)(19,22,24,23);(11,15)(14,16)(18,21)(22,23);(6,8)(17,19,20,24)(18,22,21,23);(3,4);(9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,24)(21,22);(1,2)(3,4);(5,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7783989, C₂²×#7779659
maximal normal subgroup of	C₂²×#7783989, C₂²×#7779659

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,8)(10,13)(14,16)(17,18,20,21)(19,22,24,23), (11,15)(14,16)(18,21)(22,23), (6,8)(17,19,20,24)(18,22,21,23), (3,4), (9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,24)(21,22), (1,2)(3,4), (5,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24));

C22×#46414 #72324856